Übungsblatt 4
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übung_4/agt26-blatt04.pdf
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übung_4/agt26-blatt04.pdf
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übung_4/agt_übung_3_korrpdf
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übung_4/agt_übung_4.pdf
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übung_4/agt_übung_4.tex
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übung_4/agt_übung_4.tex
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\documentclass{ngexrcs}
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\usepackage{hyperref}
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\setkeys{Gin}{pagebox=artbox, width=0.2\textwidth}
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\title{4. Übungsblatt}
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\subject{Algorithmische Graphentheorie}
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\author{Jasper Gude \and Pia Röttgers}
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\begin{document}
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\maketitle
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\points[\qquad]{20}
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\input{aufgabe_1.tex}
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\input{aufgabe_2.tex}
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\input{aufgabe_3.tex}
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\end{document}
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übung_4/aufgabe_1.tex
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übung_4/aufgabe_1.tex
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\section{Knotengrade}
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\begin{tasks}
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\item
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Einen solchen Graph gibt es. Siehe \autoref{fig:task1a}.
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\points{3}
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\begin{figure}
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\centering
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\includegraphics[page=1, width=0.8\textwidth]{figures.pdf}
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\caption{Ein Graph mit 11 Knoten und Knotengraden 1, 1, 2, 3, 4, 5, 5, 6, 7, 8, 10.}
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\label{fig:task1a}
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\end{figure}
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\item
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Das Problem kann als Maximalflussproblem modelliert werden. Dafür haben wir
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eine Quelle $s$, eine Senke $t$ und Knoten $v_i$ mit $i \in \NN_{\leq n}$.
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Die Eingangskapazität eines Knoten $v_i$ entspricht dem Eingangsgrad $e_i$.
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Die Ausgangskapazität entspricht dem Ausgangsgrad $a_i$. Siehe \autoref{fig:task1b}.
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Wenn es einen vollständigen Fluss gibt, gibt es eine Lösung für das Problem,
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sonst nicht.
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\points{4}
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\begin{figure}
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\centering
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\includegraphics[page=2, width=0.5\textwidth]{figures.pdf}
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\caption{Das Problem als Maximalflussproblem.}
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\label{fig:task1b}
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\end{figure}
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\item
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Die Modellierung ist korrekt, da der Eingangs- und Ausgangsgrad jedes Knotens $v_i$
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von oben beschränkt wird von der Eingangskapazität $e_i$ bzw. Ausgangskapazität $a_i$.
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Zudem erfüllt die Flusserhaltung den Zweck, dass die Summe der Eingangsgrade gleich der Summe der Ausgangsgrade sind, was in einem Graphen erfüllt sein muss.
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\points{2}
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\end{tasks}
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übung_4/aufgabe_2.tex
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\section{b-Flüsse}
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\begin{tasks}
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\item
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\item
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\end{tasks}
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übung_4/aufgabe_3.tex
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\section{Minimale Schnitte}
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übung_4/aufgabe_4.tex
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\section{Längste Wege}
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\begin{tasks}
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\item
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Da $s, t$ in $G'$ adjazent zu jedem Knoten in $G$ ist, können wir
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einen einfachen $s$-$t$-Weg der Länge $k+2$ erzeugen, indem wir
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einen einfachen Weg der Länge $k$ in $G$ nehmen, $s$ an das eine Ende und $t$ an das andere Ende hängen.
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Umgekehrt kann man aus einem einfachen $s$-$t$-Weg der Länge $k$
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in $G'$ einen einfachen Weg der Länge $k-2$ in $G$ konstruieren,
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indem wir $s$ und $t$ entfernen.
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\points{2}
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\item
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Ein Hamiltonweg ist ein Weg der alle Knoten in $G$ beinhaltet
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und somit Länge $n-1$ besitzt.
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Wie wir oben gezeigt haben, kann ein $s$-$t$-Weg der Länge $n+1$
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in $G'$ leicht in einen Weg der Länge $n-1$ in $G$ umgewandelt
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werden. Das heißt, dass wir einen Hamiltonweg in $G$ finden,
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wenn wir einen $s$-$t$-Weg finden.
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Umgekehrt können wir einen Hamiltonweg leicht in einen $s$-$t$-Weg
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umwandeln, also finden wir einen $s$-$t$-Weg wenn wir einen
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Hamiltonweg finden.
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Also finden wir einen Hamiltonweg genau dann, wenn wir einen
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$s$-$t$-Weg finden.
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\points{1}
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\item
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Da wir Hamiltonweg auf \algt{Längster $s$-$t$-Weg} reduziert
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haben, muss also \algt{Längster $s$-$t$-Weg} $\NPe$-schwer sein, denn
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wenn es in $\Pe$ liegen würde, könnten wir auch Hamiltonweg in
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polynomieller Zeit lösen. Da wir nicht von $\Pe = \NPe$ ausgehen,
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ist das nicht möglich.
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\points{2}
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\end{tasks}
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<color name="darkmagenta" value="0.545 0 0.545"/>
|
||||||
|
<color name="darkorange" value="1 0.549 0"/>
|
||||||
|
<color name="darkred" value="0.545 0 0"/>
|
||||||
|
<color name="gold" value="1 0.843 0"/>
|
||||||
|
<color name="gray" value="0.745"/>
|
||||||
|
<color name="green" value="0 1 0"/>
|
||||||
|
<color name="lightblue" value="0.678 0.847 0.902"/>
|
||||||
|
<color name="lightcyan" value="0.878 1 1"/>
|
||||||
|
<color name="lightgray" value="0.827"/>
|
||||||
|
<color name="lightgreen" value="0.565 0.933 0.565"/>
|
||||||
|
<color name="lightyellow" value="1 1 0.878"/>
|
||||||
|
<color name="navy" value="0 0 0.502"/>
|
||||||
|
<color name="orange" value="1 0.647 0"/>
|
||||||
|
<color name="pink" value="1 0.753 0.796"/>
|
||||||
|
<color name="purple" value="0.627 0.125 0.941"/>
|
||||||
|
<color name="red" value="1 0 0"/>
|
||||||
|
<color name="seagreen" value="0.18 0.545 0.341"/>
|
||||||
|
<color name="turquoise" value="0.251 0.878 0.816"/>
|
||||||
|
<color name="violet" value="0.933 0.51 0.933"/>
|
||||||
|
<color name="yellow" value="1 1 0"/>
|
||||||
|
<dashstyle name="dash dot dotted" value="[4 2 1 2 1 2] 0"/>
|
||||||
|
<dashstyle name="dash dotted" value="[4 2 1 2] 0"/>
|
||||||
|
<dashstyle name="dashed" value="[4] 0"/>
|
||||||
|
<dashstyle name="dotted" value="[1 3] 0"/>
|
||||||
|
<gridsize name="10 pts (~3.5 mm)" value="10"/>
|
||||||
|
<gridsize name="14 pts (~5 mm)" value="14"/>
|
||||||
|
<gridsize name="16 pts (~6 mm)" value="16"/>
|
||||||
|
<gridsize name="20 pts (~7 mm)" value="20"/>
|
||||||
|
<gridsize name="28 pts (~10 mm)" value="28"/>
|
||||||
|
<gridsize name="32 pts (~12 mm)" value="32"/>
|
||||||
|
<gridsize name="4 pts" value="4"/>
|
||||||
|
<gridsize name="56 pts (~20 mm)" value="56"/>
|
||||||
|
<gridsize name="8 pts (~3 mm)" value="8"/>
|
||||||
|
<opacity name="10%" value="0.1"/>
|
||||||
|
<opacity name="30%" value="0.3"/>
|
||||||
|
<opacity name="50%" value="0.5"/>
|
||||||
|
<opacity name="75%" value="0.75"/>
|
||||||
|
<pen name="fat" value="1.2"/>
|
||||||
|
<pen name="heavier" value="0.8"/>
|
||||||
|
<pen name="ultrafat" value="2"/>
|
||||||
|
<symbolsize name="large" value="5"/>
|
||||||
|
<symbolsize name="small" value="2"/>
|
||||||
|
<symbolsize name="tiny" value="1.1"/>
|
||||||
|
<textsize name="Huge" value="\Huge"/>
|
||||||
|
<textsize name="LARGE" value="\LARGE"/>
|
||||||
|
<textsize name="Large" value="\Large"/>
|
||||||
|
<textsize name="footnote" value="\footnotesize"/>
|
||||||
|
<textsize name="huge" value="\huge"/>
|
||||||
|
<textsize name="large" value="\large"/>
|
||||||
|
<textsize name="script" value="\scriptsize"/>
|
||||||
|
<textsize name="small" value="\small"/>
|
||||||
|
<textsize name="tiny" value="\tiny"/>
|
||||||
|
<textstyle name="center" begin="\begin{center}" end="\end{center}"/>
|
||||||
|
<textstyle name="item" begin="\begin{itemize}\item{}" end="\end{itemize}"/>
|
||||||
|
<textstyle name="itemize" begin="\begin{itemize}" end="\end{itemize}"/>
|
||||||
|
<tiling name="falling" angle="-60" step="4" width="1"/>
|
||||||
|
<tiling name="rising" angle="30" step="4" width="1"/>
|
||||||
|
</ipestyle>
|
||||||
|
<ipestyle name="ngslides">
|
||||||
|
<symbol name="arrow/ng-fnormal(spx)">
|
||||||
|
<path stroke="sym-stroke" fill="white" pen="sym-pen">
|
||||||
|
-0.7 0 m
|
||||||
|
-1.7 0.333 l
|
||||||
|
-1.7 -0.333 l
|
||||||
|
h
|
||||||
|
</path>
|
||||||
|
</symbol>
|
||||||
|
<symbol name="arrow/ng-normal(spx)">
|
||||||
|
<path stroke="sym-stroke" fill="sym-stroke" pen="sym-pen">
|
||||||
|
-0.7 0 m
|
||||||
|
-1.7 0.333 l
|
||||||
|
-1.7 -0.333 l
|
||||||
|
h
|
||||||
|
</path>
|
||||||
|
</symbol>
|
||||||
|
<symbol name="arrow/ng-pointed(spx)">
|
||||||
|
<path stroke="sym-stroke" fill="sym-stroke" pen="sym-pen">
|
||||||
|
-0.7 0 m
|
||||||
|
-1.7 0.333 l
|
||||||
|
-0.8 0 l
|
||||||
|
-1.7 -0.333 l
|
||||||
|
h
|
||||||
|
</path>
|
||||||
|
</symbol>
|
||||||
|
<symbol name="arrow/ng-fpointed(spx)">
|
||||||
|
<path stroke="sym-stroke" fill="white" pen="sym-pen">
|
||||||
|
-0.7 0 m
|
||||||
|
-1.7 0.333 l
|
||||||
|
-0.8 0 l
|
||||||
|
-1.7 -0.333 l
|
||||||
|
h
|
||||||
|
</path>
|
||||||
|
</symbol>
|
||||||
|
<symbol name="arrow/ng-linear(spx)">
|
||||||
|
<path stroke="sym-stroke" pen="sym-pen">
|
||||||
|
-1.7 0.333 m
|
||||||
|
-0.7 0 l
|
||||||
|
-1.7 -0.333 l
|
||||||
|
</path>
|
||||||
|
</symbol>
|
||||||
|
<symbol name="arrow/ng-fdouble(spx)">
|
||||||
|
<path stroke="sym-stroke" fill="white" pen="sym-pen">
|
||||||
|
-0.7 0 m
|
||||||
|
-1.7 0.333 l
|
||||||
|
-1.7 -0.333 l
|
||||||
|
h
|
||||||
|
-1.7 0 m
|
||||||
|
-2.7 0.333 l
|
||||||
|
-2.7 -0.333 l
|
||||||
|
h
|
||||||
|
</path>
|
||||||
|
</symbol>
|
||||||
|
<symbol name="arrow/ng-double(spx)">
|
||||||
|
<path stroke="sym-stroke" fill="sym-stroke" pen="sym-pen">
|
||||||
|
-0.7 0 m
|
||||||
|
-1.7 0.333 l
|
||||||
|
-1.7 -0.333 l
|
||||||
|
h
|
||||||
|
-1.7 0 m
|
||||||
|
-2.7 0.333 l
|
||||||
|
-2.7 -0.333 l
|
||||||
|
h
|
||||||
|
</path>
|
||||||
|
</symbol>
|
||||||
|
<symbol name="decoration/remark">
|
||||||
|
<path stroke="dark gray" fill="light yellow">
|
||||||
|
100 204 m
|
||||||
|
4 0 0 4 100 200 96 200 a
|
||||||
|
96 100 l
|
||||||
|
4 0 0 4 100 100 100 96 a
|
||||||
|
300 96 l
|
||||||
|
4 0 0 4 300 100 304 100 a
|
||||||
|
304 200 l
|
||||||
|
4 0 0 4 300 200 300 204 a
|
||||||
|
h
|
||||||
|
</path>
|
||||||
|
</symbol>
|
||||||
|
<symbol name="decoration/theorem">
|
||||||
|
<path stroke="dark blue" fill="light blue">
|
||||||
|
100 204 m
|
||||||
|
4 0 0 4 100 200 96 200 a
|
||||||
|
96 100 l
|
||||||
|
4 0 0 4 100 100 100 96 a
|
||||||
|
300 96 l
|
||||||
|
4 0 0 4 300 100 304 100 a
|
||||||
|
304 200 l
|
||||||
|
4 0 0 4 300 200 300 204 a
|
||||||
|
h
|
||||||
|
</path>
|
||||||
|
</symbol>
|
||||||
|
<symbol name="decoration/problem">
|
||||||
|
<path stroke="dark green" fill="light green">
|
||||||
|
100 204 m
|
||||||
|
4 0 0 4 100 200 96 200 a
|
||||||
|
96 100 l
|
||||||
|
4 0 0 4 100 100 100 96 a
|
||||||
|
300 96 l
|
||||||
|
4 0 0 4 300 100 304 100 a
|
||||||
|
304 200 l
|
||||||
|
4 0 0 4 300 200 300 204 a
|
||||||
|
h
|
||||||
|
</path>
|
||||||
|
</symbol>
|
||||||
|
<symbol name="decoration/definition">
|
||||||
|
<path stroke="dark gray" fill="light yellow">
|
||||||
|
96 96 m
|
||||||
|
304 96 l
|
||||||
|
304 204 l
|
||||||
|
96 204 l
|
||||||
|
h
|
||||||
|
</path>
|
||||||
|
</symbol>
|
||||||
|
<symbol name="decoration/algorithm">
|
||||||
|
<path stroke="dark gray" fill="light green" dash="dashed">
|
||||||
|
96 96 m
|
||||||
|
304 96 l
|
||||||
|
304 204 l
|
||||||
|
96 204 l
|
||||||
|
h
|
||||||
|
</path>
|
||||||
|
</symbol>
|
||||||
|
<symbol name="decoration/graph">
|
||||||
|
<path stroke="dark gray">
|
||||||
|
92 92 m
|
||||||
|
308 92 l
|
||||||
|
308 208 l
|
||||||
|
92 208 l
|
||||||
|
h
|
||||||
|
</path>
|
||||||
|
</symbol>
|
||||||
|
<symbol name="decoration/comic-bl">
|
||||||
|
<path stroke="dark gray" fill="light yellow">
|
||||||
|
100 96 m
|
||||||
|
110 96 s
|
||||||
|
110 92
|
||||||
|
106 88 s
|
||||||
|
116 92
|
||||||
|
118 96 s
|
||||||
|
300 96 l
|
||||||
|
4 0 0 4 300 100 304 100 a
|
||||||
|
304 200 l
|
||||||
|
4 0 0 4 300 200 300 204 a
|
||||||
|
100 204 l
|
||||||
|
4 0 0 4 100 200 96 200 a
|
||||||
|
96 100 l
|
||||||
|
4 0 0 4 100 100 100 96 a
|
||||||
|
h
|
||||||
|
</path>
|
||||||
|
</symbol>
|
||||||
|
<symbol name="decoration/comic-bm">
|
||||||
|
<path stroke="dark gray" fill="light yellow">
|
||||||
|
100 96 m
|
||||||
|
186 96 s
|
||||||
|
186 92
|
||||||
|
182 88 s
|
||||||
|
192 92
|
||||||
|
194 96 s
|
||||||
|
300 96 l
|
||||||
|
4 0 0 4 300 100 304 100 a
|
||||||
|
304 200 l
|
||||||
|
4 0 0 4 300 200 300 204 a
|
||||||
|
100 204 l
|
||||||
|
4 0 0 4 100 200 96 200 a
|
||||||
|
96 100 l
|
||||||
|
4 0 0 4 100 100 100 96 a
|
||||||
|
h
|
||||||
|
</path>
|
||||||
|
</symbol>
|
||||||
|
<symbol name="decoration/comic-br">
|
||||||
|
<path matrix="-1 0 0 1 400 0" stroke="dark gray" fill="light yellow">
|
||||||
|
100 96 m
|
||||||
|
110 96 s
|
||||||
|
110 92
|
||||||
|
106 88 s
|
||||||
|
116 92
|
||||||
|
118 96 s
|
||||||
|
300 96 l
|
||||||
|
4 0 0 4 300 100 304 100 a
|
||||||
|
304 200 l
|
||||||
|
4 0 0 4 300 200 300 204 a
|
||||||
|
100 204 l
|
||||||
|
4 0 0 4 100 200 96 200 a
|
||||||
|
96 100 l
|
||||||
|
4 0 0 4 100 100 100 96 a
|
||||||
|
h
|
||||||
|
</path>
|
||||||
|
</symbol>
|
||||||
|
<symbol name="decoration/comic-tl">
|
||||||
|
<path matrix="1 0 0 -1 0 300" stroke="dark gray" fill="light yellow">
|
||||||
|
100 96 m
|
||||||
|
110 96 s
|
||||||
|
110 92
|
||||||
|
106 88 s
|
||||||
|
116 92
|
||||||
|
118 96 s
|
||||||
|
300 96 l
|
||||||
|
4 0 0 4 300 100 304 100 a
|
||||||
|
304 200 l
|
||||||
|
4 0 0 4 300 200 300 204 a
|
||||||
|
100 204 l
|
||||||
|
4 0 0 4 100 200 96 200 a
|
||||||
|
96 100 l
|
||||||
|
4 0 0 4 100 100 100 96 a
|
||||||
|
h
|
||||||
|
</path>
|
||||||
|
</symbol>
|
||||||
|
<symbol name="decoration/comic-tm">
|
||||||
|
<path matrix="1 0 0 -1 0 300" stroke="dark gray" fill="light yellow">
|
||||||
|
100 96 m
|
||||||
|
186 96 s
|
||||||
|
186 92
|
||||||
|
182 88 s
|
||||||
|
192 92
|
||||||
|
194 96 s
|
||||||
|
300 96 l
|
||||||
|
4 0 0 4 300 100 304 100 a
|
||||||
|
304 200 l
|
||||||
|
4 0 0 4 300 200 300 204 a
|
||||||
|
100 204 l
|
||||||
|
4 0 0 4 100 200 96 200 a
|
||||||
|
96 100 l
|
||||||
|
4 0 0 4 100 100 100 96 a
|
||||||
|
h
|
||||||
|
</path>
|
||||||
|
</symbol>
|
||||||
|
<symbol name="decoration/comic-tr">
|
||||||
|
<path matrix="-1 0 0 -1 400 300" stroke="dark gray" fill="light yellow">
|
||||||
|
100 96 m
|
||||||
|
110 96 s
|
||||||
|
110 92
|
||||||
|
106 88 s
|
||||||
|
116 92
|
||||||
|
118 96 s
|
||||||
|
300 96 l
|
||||||
|
4 0 0 4 300 100 304 100 a
|
||||||
|
304 200 l
|
||||||
|
4 0 0 4 300 200 300 204 a
|
||||||
|
100 204 l
|
||||||
|
4 0 0 4 100 200 96 200 a
|
||||||
|
96 100 l
|
||||||
|
4 0 0 4 100 100 100 96 a
|
||||||
|
h
|
||||||
|
</path>
|
||||||
|
</symbol>
|
||||||
|
<color name="dark blue" value="0.121 0.47 0.705"/>
|
||||||
|
<color name="dark brown" value="0.651 0.337 0.157"/>
|
||||||
|
<color name="dark cyan" value="0.106 0.62 0.467"/>
|
||||||
|
<color name="dark gray" value="0.5"/>
|
||||||
|
<color name="dark green" value="0.2 0.627 0.172"/>
|
||||||
|
<color name="dark orange" value="1 0.498 0"/>
|
||||||
|
<color name="dark pink" value="0.969 0.506 0.749"/>
|
||||||
|
<color name="dark purple" value="0.415 0.239 0.603"/>
|
||||||
|
<color name="dark red" value="0.89 0.102 0.109"/>
|
||||||
|
<color name="dark yellow" value="1 1 0.2"/>
|
||||||
|
<color name="light blue" value="0.651 0.807 0.89"/>
|
||||||
|
<color name="light brown" value="0.898 0.847 0.741"/>
|
||||||
|
<color name="light cyan" value="0.553 0.827 0.78"/>
|
||||||
|
<color name="light gray" value="0.8"/>
|
||||||
|
<color name="light green" value="0.698 0.874 0.541"/>
|
||||||
|
<color name="light orange" value="0.992 0.749 0.435"/>
|
||||||
|
<color name="light pink" value="0.992 0.855 0.925"/>
|
||||||
|
<color name="light purple" value="0.792 0.698 0.839"/>
|
||||||
|
<color name="light red" value="0.984 0.603 0.6"/>
|
||||||
|
<color name="light yellow" value="1 1 0.8"/>
|
||||||
|
<color name="pagenumber" value="0.5"/>
|
||||||
|
<color name="title" value="0.247 0.282 0.392"/>
|
||||||
|
<pen name="fat" value="3.6"/>
|
||||||
|
<pen name="heavier" value="2.4"/>
|
||||||
|
<pen name="normal" value="1.2"/>
|
||||||
|
<pen name="ultrafat" value="6"/>
|
||||||
|
<symbolsize name="large" value="15"/>
|
||||||
|
<symbolsize name="normal" value="9"/>
|
||||||
|
<symbolsize name="small" value="6"/>
|
||||||
|
<symbolsize name="tiny" value="3.3"/>
|
||||||
|
<textstretch name="Huge" value="2.4"/>
|
||||||
|
<textstretch name="LARGE" value="2.4"/>
|
||||||
|
<textstretch name="Large" value="2.4"/>
|
||||||
|
<textstretch name="footnote" value="2.4"/>
|
||||||
|
<textstretch name="huge" value="2.4"/>
|
||||||
|
<textstretch name="large" value="2.4"/>
|
||||||
|
<textstretch name="normal" value="2.4"/>
|
||||||
|
<textstretch name="script" value="2.4"/>
|
||||||
|
<textstretch name="small" value="2.4"/>
|
||||||
|
<textstretch name="tiny" value="2.4"/>
|
||||||
|
<textstyle name="algorithmic" begin="\begin{algorithmic}" end="\end{algorithmic}"/>
|
||||||
|
<textstyle name="definition" begin="\begin{definition}" end="\end{definition}"/>
|
||||||
|
<textstyle name="enumerate" begin="\begin{enumerate}" end="\end{enumerate}"/>
|
||||||
|
<textstyle name="example" begin="\begin{example}" end="\end{example}"/>
|
||||||
|
<textstyle name="lemma" begin="\begin{lemma}" end="\end{lemma}"/>
|
||||||
|
<textstyle name="proof" begin="\begin{proof}" end="\end{proof}"/>
|
||||||
|
<textstyle name="theorem" begin="\begin{theorem}" end="\end{theorem}"/>
|
||||||
|
<preamble>
|
||||||
|
\usepackage[ngerman]{babel}
|
||||||
|
\usepackage[sansdefault]{fontsetup}
|
||||||
|
\usepackage{ngutils}
|
||||||
|
|
||||||
|
\renewcommand{\theoremfont}{\normalfont\bfseries\color{dark blue}}
|
||||||
|
\renewcommand{\definitionfont}{\normalfont\bfseries\color{dark gray}}
|
||||||
|
|
||||||
|
\newcommand{\labelfont}{\normalfont\bfseries\color{dark blue}}
|
||||||
|
|
||||||
|
%\setlength{\leftmargini}{1.33em}
|
||||||
|
%\setlength{\leftmarginii}{1.33em}
|
||||||
|
%\setlength{\leftmarginiii}{1.33em}
|
||||||
|
%\setlength{\leftmarginvi}{1.33em}
|
||||||
|
|
||||||
|
\renewcommand{\labelenumi}{\labelfont\arabic{enumi}.}
|
||||||
|
\renewcommand{\labelenumii}{\labelfont\alph{enumii})}
|
||||||
|
\renewcommand{\labelenumiii}{\labelfont\roman{enumiii}.}
|
||||||
|
\renewcommand{\labelenumiv}{\labelfont\Alph{enumiv}.}
|
||||||
|
|
||||||
|
\renewcommand{\labelitemi}{\labelfont $\blockfull$}
|
||||||
|
\renewcommand{\labelitemii}{\labelfont $\blackpointerright$}
|
||||||
|
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|
||||||
|
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|
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|
|
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|
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|
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|
|
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|
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|
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|
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|
||||||
|
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|
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|
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|
||||||
|
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|
||||||
|
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|
||||||
|
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|
||||||
|
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|
||||||
|
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|
||||||
|
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|
||||||
|
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|
||||||
|
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|
||||||
|
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|
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|
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|
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384 384 m
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192 128 l
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<path stroke="black">
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384 384 m
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576 128 l
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<path stroke="black">
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384 384 m
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192 384 l
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<path stroke="black">
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384 384 m
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480 352 l
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<path stroke="black">
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384 384 m
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576 384 l
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480 160 m
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192 256 l
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<path stroke="black">
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480 160 m
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384 128 l
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<path stroke="black">
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480 160 m
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192 128 l
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<path stroke="black">
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480 160 m
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576 128 l
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</path>
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<path stroke="black">
|
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480 160 m
|
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192 384 l
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</path>
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<path stroke="black">
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480 160 m
|
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|
480 352 l
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|
</path>
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<path stroke="black">
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480 160 m
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|
576 384 l
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|
</path>
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<path stroke="black">
|
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|
384 128 m
|
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|
192 128 l
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|
</path>
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<path stroke="black">
|
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|
384 128 m
|
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|
576 128 l
|
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</path>
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<path stroke="black">
|
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|
384 128 m
|
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|
192 384 l
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|
</path>
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<path stroke="black">
|
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|
384 128 m
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|
480 352 l
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|
</path>
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|
<path stroke="black">
|
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|
384 128 m
|
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|
576 384 l
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|
</path>
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|
<path stroke="black">
|
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|
576 128 m
|
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|
192 384 l
|
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|
</path>
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|
<path stroke="black">
|
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|
576 128 m
|
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|
480 352 l
|
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|
</path>
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<path stroke="black">
|
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|
576 128 m
|
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|
576 384 l
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|
</path>
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<path stroke="black">
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480 352 m
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576 384 l
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|
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|
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416 256 m
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416 448 l
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<path stroke="black">
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416 256 m
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|
320 416 l
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<path stroke="black">
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|
416 256 m
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|
416 64 l
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|
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<path stroke="black">
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|
416 256 m
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|
256 352 l
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|
</path>
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<path stroke="black">
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|
416 256 m
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|
560 112 l
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|
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|
<path stroke="black">
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|
416 256 m
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|
224 256 l
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|
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<path stroke="black">
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|
416 256 m
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|
608 256 l
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|
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<path stroke="black">
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|
416 256 m
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|
272 112 l
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|
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|
<path stroke="black">
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|
416 256 m
|
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|
512 416 l
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|
</path>
|
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|
<path stroke="black">
|
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|
416 256 m
|
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|
576 352 l
|
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|
</path>
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|
<path stroke="black">
|
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|
416 64 m
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|
256 352 l
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|
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|
<path stroke="black">
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|
416 64 m
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|
560 112 l
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|
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|
<path stroke="black">
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|
416 64 m
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|
224 256 l
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|
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|
<path stroke="black">
|
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|
416 64 m
|
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|
608 256 l
|
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|
</path>
|
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|
<path stroke="black">
|
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|
416 64 m
|
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|
272 112 l
|
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|
</path>
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|
<path stroke="black">
|
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|
416 64 m
|
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|
512 416 l
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|
</path>
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|
<path stroke="black">
|
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|
416 64 m
|
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|
576 352 l
|
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|
</path>
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|
<path stroke="black">
|
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|
560 112 m
|
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|
224 256 l
|
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|
</path>
|
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|
<path stroke="black">
|
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|
560 112 m
|
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|
608 256 l
|
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|
</path>
|
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|
<path stroke="black">
|
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|
560 112 m
|
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|
272 112 l
|
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|
</path>
|
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|
<path stroke="black">
|
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|
560 112 m
|
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|
512 416 l
|
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|
</path>
|
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|
<path stroke="black">
|
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|
560 112 m
|
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|
576 352 l
|
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|
</path>
|
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|
<path stroke="black">
|
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|
608 256 m
|
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|
272 112 l
|
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|
</path>
|
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|
<path stroke="black">
|
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|
608 256 m
|
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|
512 416 l
|
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|
</path>
|
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|
<path stroke="black">
|
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|
608 256 m
|
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|
576 352 l
|
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|
</path>
|
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|
<path stroke="black">
|
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|
512 416 m
|
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|
576 352 l
|
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|
</path>
|
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|
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|
||||||
|
<text matrix="1 0 0 1 -160 -312" transformations="translations" pos="352 448" stroke="black" type="label" halign="center" valign="bottom" size="script" style="math">3</text>
|
||||||
|
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|
||||||
|
<text matrix="1 0 0 1 128 -88" transformations="translations" pos="352 448" stroke="black" type="label" halign="center" valign="bottom" size="script" style="math">5</text>
|
||||||
|
<text matrix="1 0 0 1 224 -56" transformations="translations" pos="352 448" stroke="black" type="label" halign="center" valign="bottom" size="script" style="math">5</text>
|
||||||
|
<text matrix="1 0 0 1 128 -296" transformations="translations" pos="352 448" stroke="black" type="label" halign="center" valign="top" size="script" style="math">8</text>
|
||||||
|
<text matrix="1 0 0 1 272 -184" transformations="translations" pos="352 448" stroke="black" type="label" halign="center" valign="top" size="script" style="math">7</text>
|
||||||
|
<text matrix="1 0 0 1 288 -320" transformations="translations" pos="352 448" stroke="black" type="label" halign="center" valign="top" size="script" style="math">6</text>
|
||||||
|
<text matrix="1 0 0 1 32 -56" transformations="translations" pos="352 448" stroke="black" type="label" halign="center" valign="bottom" size="script" style="math">10</text>
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|
<path stroke="black">
|
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|
416 256 m
|
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|
416 64 l
|
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|
</path>
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|
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<page>
|
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|
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|
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|
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|
448 320 m
|
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|
576 256 l
|
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|
</path>
|
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|
<path stroke="black" arrow="ng-normal/normal">
|
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|
448 256 m
|
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|
576 256 l
|
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|
</path>
|
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|
<path stroke="black" arrow="ng-normal/normal">
|
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|
448 192 m
|
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|
576 256 l
|
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|
</path>
|
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|
<path stroke="black" arrow="ng-normal/normal">
|
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|
192 256 m
|
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|
320 320 l
|
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|
</path>
|
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|
<path stroke="black" arrow="ng-normal/normal">
|
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|
192 256 m
|
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|
320 256 l
|
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|
</path>
|
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|
<path stroke="black" arrow="ng-normal/normal">
|
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|
192 256 m
|
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|
320 192 l
|
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|
</path>
|
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|
<path stroke="black" arrow="ng-normal/normal">
|
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|
320 256 m
|
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|
448 320 l
|
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|
</path>
|
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|
<path stroke="black" arrow="ng-normal/normal">
|
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|
320 320 m
|
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|
448 320 l
|
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|
</path>
|
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|
<path stroke="black" arrow="ng-normal/normal">
|
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|
320 320 m
|
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|
448 320 l
|
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|
</path>
|
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|
<path stroke="black" arrow="ng-normal/normal">
|
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|
320 256 m
|
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|
448 256 l
|
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|
</path>
|
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|
<path stroke="black" arrow="ng-normal/normal">
|
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|
320 320 m
|
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|
448 192 l
|
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|
</path>
|
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|
<path stroke="black" arrow="ng-normal/normal">
|
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|
320 192 m
|
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|
448 192 l
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|
</path>
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|
<path stroke="black" arrow="ng-normal/normal">
|
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|
320 320 m
|
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|
448 256 l
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||||||
|
<use name="mark/disk(sx)" pos="448 192" size="normal" stroke="dark orange"/>
|
||||||
|
<use name="mark/disk(sx)" pos="576 256" size="normal" stroke="black"/>
|
||||||
|
<text matrix="1 0 0 1 0 8" transformations="translations" pos="320 320" stroke="black" type="label" width="18.7176" height="8.5848" depth="4.728" halign="center" valign="bottom" size="footnote" style="math">v_1</text>
|
||||||
|
<text matrix="1 0 0 1 0 -56" transformations="translations" pos="320 320" stroke="black" type="label" width="18.7176" height="8.5848" depth="4.728" halign="center" valign="bottom" size="footnote" style="math">v_2</text>
|
||||||
|
<text matrix="1 0 0 1 4 -136" transformations="translations" pos="320 320" stroke="black" type="label" width="19.764" height="8.604" depth="4.824" halign="center" valign="top" size="footnote" style="math">v_n</text>
|
||||||
|
<text matrix="1 0 0 1 128 8" transformations="translations" pos="320 320" stroke="black" type="label" width="17.9904" height="8.8152" depth="4.728" halign="center" valign="bottom" size="footnote" style="math">e_1</text>
|
||||||
|
<text matrix="1 0 0 1 -128 -56" transformations="translations" pos="320 320" stroke="black" type="label" width="7.7664" height="8.8368" depth="0.192" halign="center" valign="bottom" size="footnote" style="math">s</text>
|
||||||
|
<text matrix="1 0 0 1 128 -56" transformations="translations" pos="320 320" stroke="black" type="label" width="17.9904" height="8.8152" depth="4.728" halign="center" valign="bottom" size="footnote" style="math">e_2</text>
|
||||||
|
<text matrix="1 0 0 1 128 -136" transformations="translations" pos="320 320" stroke="black" type="label" width="21.6624" height="8.8344" depth="4.824" halign="center" valign="top" size="footnote" style="math">e_m</text>
|
||||||
|
<text matrix="1 0 0 1 256 -56" transformations="translations" pos="320 320" stroke="black" type="label" width="8.34" height="10.9416" depth="0.192" halign="center" valign="bottom" size="footnote" style="math">t</text>
|
||||||
|
<text matrix="0.894704 0.446659 -0.446659 0.894704 112.622 -137.236" pos="320 320" stroke="black" type="label" width="33.0312" height="8.976" depth="3.984" halign="center" valign="bottom" size="tiny" style="math">B(v_1)</text>
|
||||||
|
<text matrix="0.894427 -0.447214 0.447214 0.894427 -73.1492 134.135" pos="256 224" stroke="black" type="label" width="33.9288" height="8.9664" depth="3.048" halign="center" valign="top" size="tiny" style="math">B(v_n)</text>
|
||||||
|
<text transformations="translations" pos="256 260" stroke="black" type="label" width="33.0312" height="8.976" depth="3.984" halign="center" valign="bottom" size="tiny" style="math">B(v_2)</text>
|
||||||
|
<text matrix="0.894427 -0.447214 0.447214 0.894427 186.851 218.135" pos="256 224" stroke="black" type="label" width="28.2504" height="8.976" depth="3.984" halign="center" valign="top" size="tiny" style="math">r(e_1)</text>
|
||||||
|
<text matrix="0.894704 0.446659 -0.446659 0.894704 368.622 -209.236" pos="320 320" stroke="black" type="label" width="31.6704" height="8.9664" depth="3.048" halign="center" valign="top" size="tiny" style="math">r(e_m)</text>
|
||||||
|
<text matrix="1 0 0 1 256 0" transformations="translations" pos="256 260" stroke="black" type="label" width="28.2504" height="8.976" depth="3.984" halign="center" valign="bottom" size="tiny" style="math">r(e_2)</text>
|
||||||
|
<text transformations="translations" pos="320 224" stroke="black" type="label" width="5.3184" height="10.98" depth="1.32" halign="center" valign="center" size="footnote" style="math">\vdots</text>
|
||||||
|
<text matrix="1 0 0 1 128 0" transformations="translations" pos="320 224" stroke="black" type="label" width="5.3184" height="10.98" depth="1.32" halign="center" valign="center" size="footnote" style="math">\vdots</text>
|
||||||
|
</page>
|
||||||
|
</ipe>
|
||||||
71
übung_4/ngexrcs.cls
Normal file
71
übung_4/ngexrcs.cls
Normal file
|
|
@ -0,0 +1,71 @@
|
||||||
|
% vim: set filetype:tex
|
||||||
|
|
||||||
|
% Identification %
|
||||||
|
\NeedsTeXFormat{LaTeX2e}
|
||||||
|
\ProvidesClass{ngexrcs}[2021/12/21 ADS Exercise class]
|
||||||
|
|
||||||
|
% Handle options %
|
||||||
|
\DeclareOption*{\PassOptionsToClass{\CurrentOption}{scrartcl}}
|
||||||
|
\ProcessOptions\relax
|
||||||
|
|
||||||
|
% More declarations %
|
||||||
|
\LoadClass{scrartcl}
|
||||||
|
|
||||||
|
\KOMAoptions{mpinclude=true}
|
||||||
|
\recalctypearea
|
||||||
|
|
||||||
|
\RequirePackage{ngutils}
|
||||||
|
|
||||||
|
% define fonts
|
||||||
|
\RequirePackage{fontspec}
|
||||||
|
\RequirePackage[math-style=upright]{unicode-math}
|
||||||
|
\setmainfont{TeX Gyre Pagella}
|
||||||
|
\setsansfont{TeX Gyre Heros}
|
||||||
|
\setmonofont{TeX Gyre Cursor}
|
||||||
|
\setmathfont{Euler Math}
|
||||||
|
|
||||||
|
\setlength{\marginparwidth}{1.5\marginparwidth}
|
||||||
|
\setlength{\fboxrule}{\heavyrulewidth}
|
||||||
|
|
||||||
|
% Use sansfont for all title elements
|
||||||
|
\addtokomafont{titlehead}{\sffamily}
|
||||||
|
\addtokomafont{subject}{\normalfont \sffamily \itshape}
|
||||||
|
\addtokomafont{title}{\sffamily}
|
||||||
|
\addtokomafont{subtitle}{\sffamily}
|
||||||
|
\addtokomafont{author}{\sffamily}
|
||||||
|
\addtokomafont{date}{\sffamily}
|
||||||
|
\addtokomafont{publishers}{\sffamily}
|
||||||
|
|
||||||
|
% Use serif font for headings
|
||||||
|
\addtokomafont{disposition}{\rmfamily}
|
||||||
|
|
||||||
|
% Let sections be formated as in: Aufgabe 1 -- Section title
|
||||||
|
\renewcommand*{\sectionformat}{Aufgabe \thesection\autodot\enskip--\enskip}
|
||||||
|
|
||||||
|
% Let points of an exercise be printed as in: [__ / 2]
|
||||||
|
\newkomafont{points}{\sffamily}
|
||||||
|
\newcommand\points[2][\quad]{\marginline{\framebox{{\usekomafont{points}#1 \textbf{/} #2}}}}
|
||||||
|
|
||||||
|
\newcounter{task}
|
||||||
|
\renewcommand{\thetask}{\@alph\c@task)}
|
||||||
|
\newenvironment{tasks}
|
||||||
|
{
|
||||||
|
\begin{list}{\thetask}
|
||||||
|
{
|
||||||
|
\usecounter{task}
|
||||||
|
\setlength{\leftmargin}{1.6em}
|
||||||
|
}
|
||||||
|
}{%
|
||||||
|
\end{list}
|
||||||
|
}
|
||||||
|
|
||||||
|
\DeclareNewTOC
|
||||||
|
[
|
||||||
|
type=algorithm,
|
||||||
|
types=algorithms,
|
||||||
|
float,
|
||||||
|
floattype=4,
|
||||||
|
name=Algorithmus,
|
||||||
|
listname={Algorithmenverzeichnis}
|
||||||
|
]
|
||||||
|
{loa}
|
||||||
124
übung_4/ngutils.sty
Normal file
124
übung_4/ngutils.sty
Normal file
|
|
@ -0,0 +1,124 @@
|
||||||
|
\NeedsTeXFormat{LaTeX2e}
|
||||||
|
\ProvidesPackage{ngutils}[2025/12/24 Never's LaTeX utils]
|
||||||
|
|
||||||
|
\RequirePackage[ngerman]{babel}
|
||||||
|
\RequirePackage{graphicx}
|
||||||
|
\RequirePackage{tabularx}
|
||||||
|
\RequirePackage{booktabs}
|
||||||
|
\RequirePackage{listings}
|
||||||
|
|
||||||
|
\lstnewenvironment{pseudocode}[1][] %defines the algorithm listing environment
|
||||||
|
{
|
||||||
|
\renewcommand{\lstlistingname}{Algorithmus}
|
||||||
|
\lstset{ %this is the stype
|
||||||
|
mathescape=true,
|
||||||
|
columns=fullflexible,
|
||||||
|
basicstyle=\normalfont,
|
||||||
|
identifierstyle=\scshape,
|
||||||
|
keywordstyle=\bfseries,
|
||||||
|
keywords={, new, if, then, else, while, for, in, to, up, down, foreach, do, return},
|
||||||
|
commentstyle=\itshape,
|
||||||
|
comment=[l]//,
|
||||||
|
delim=[is][\normalfont]{|}{|},
|
||||||
|
tabsize=3,
|
||||||
|
frame=tb,
|
||||||
|
framerule=1pt,
|
||||||
|
#1 % this is to add specific settings to an usage of this environment (for instnce, the caption and referable label)
|
||||||
|
}
|
||||||
|
}
|
||||||
|
{}
|
||||||
|
|
||||||
|
\newcommand{\seilpmi}{\Longleftarrow}
|
||||||
|
|
||||||
|
\newcommand{\NN}{\mathbb{N}}
|
||||||
|
\newcommand{\ZZ}{\mathbb{Z}}
|
||||||
|
\newcommand{\QQ}{\mathbb{Q}}
|
||||||
|
\newcommand{\RR}{\mathbb{R}}
|
||||||
|
\newcommand{\CC}{\mathbb{C}}
|
||||||
|
\newcommand{\PP}{\mathbb{P}}
|
||||||
|
|
||||||
|
\newcommand{\Pe}{\mathrm{P}}
|
||||||
|
\newcommand{\NPe}{\mathrm{NP}}
|
||||||
|
|
||||||
|
\newcommand{\Oh}{\mathcal{O}}
|
||||||
|
\newcommand{\oh}{\scriptstyle{\mathcal{O}}}
|
||||||
|
|
||||||
|
\newcommand{\Eh}{\mathop{\mathrm{E}}}
|
||||||
|
\newcommand{\Var}{\mathop{\mathrm{Var}}}
|
||||||
|
\newcommand{\pot}{\mathop{\mathcal{P}}}
|
||||||
|
\newcommand{\argmax}{\mathop{\mathrm{arg\,max}}}
|
||||||
|
\newcommand{\argmin}{\mathop{\mathrm{arg\,min}}}
|
||||||
|
|
||||||
|
\newcommand{\parens}[1]{\left(#1\right)}
|
||||||
|
\newcommand{\brackets}[1]{\left[#1\right]}
|
||||||
|
\newcommand{\braces}[1]{\left\{#1\right\}}
|
||||||
|
\newcommand{\angled}[1]{\left\langle#1\right\rangle}
|
||||||
|
|
||||||
|
\newcommand{\tup}[1]{\parens{#1}}
|
||||||
|
\newcommand{\ivl}[1]{\brackets{#1}}
|
||||||
|
\newcommand{\set}[1]{\braces{#1}}
|
||||||
|
\newcommand{\arr}[1]{\angled{#1}}
|
||||||
|
\newcommand{\abs}[1]{\lvert#1\rvert}
|
||||||
|
\newcommand{\norm}[1]{\lVert#1\rVert}
|
||||||
|
\newcommand{\ceil}[1]{\lceil#1\rceil}
|
||||||
|
\newcommand{\floor}[1]{\lfloor#1\rfloor}
|
||||||
|
|
||||||
|
\newcommand{\alg}[2]{{\normalfont\scshape#1}{\normalfont(#2)}}
|
||||||
|
\newcommand{\algt}[1]{{\normalfont\scshape#1}}
|
||||||
|
\newcommand{\com}[1]{{\normalfont\itshape/\!\!/ #1}}
|
||||||
|
|
||||||
|
\newenvironment{algorithmic}[1][1em]{
|
||||||
|
\begin{minipage}[t]{\dimexpr\linewidth}
|
||||||
|
\bfseries
|
||||||
|
\begin{tabbing}
|
||||||
|
\hspace{#1}\=\hspace{#1}\=\hspace{#1}\=\hspace{#1}\=\hspace{#1}\=\hspace{#1}\=\hspace{#1}\=\hspace{#1}\=\kill
|
||||||
|
}{%
|
||||||
|
\end{tabbing}
|
||||||
|
\end{minipage}
|
||||||
|
}
|
||||||
|
|
||||||
|
\newlength{\theoremskip}
|
||||||
|
\setlength{\theoremskip}{6em}
|
||||||
|
|
||||||
|
\newcommand{\theoremfont}{\normalfont\bfseries}
|
||||||
|
\newcommand{\definitionfont}{\normalfont\bfseries}
|
||||||
|
|
||||||
|
\newenvironment{theoremic}[2][\theoremfont]{
|
||||||
|
{#1 #2}
|
||||||
|
\hfill
|
||||||
|
\begin{minipage}[t]{\the\dimexpr\linewidth-\theoremskip\relax}
|
||||||
|
}{
|
||||||
|
\end{minipage}
|
||||||
|
}
|
||||||
|
|
||||||
|
\newenvironment{theorem}{
|
||||||
|
\begin{theoremic}{Satz.}
|
||||||
|
}{
|
||||||
|
\end{theoremic}
|
||||||
|
}
|
||||||
|
|
||||||
|
\newenvironment{lemma}{
|
||||||
|
\begin{theoremic}{Lemma.}
|
||||||
|
}{
|
||||||
|
\end{theoremic}
|
||||||
|
}
|
||||||
|
|
||||||
|
\newenvironment{proof}{
|
||||||
|
\begin{theoremic}{Beweis.}
|
||||||
|
}{
|
||||||
|
\end{theoremic}
|
||||||
|
}
|
||||||
|
|
||||||
|
\newenvironment{definition}{
|
||||||
|
\begin{theoremic}[\definitionfont]{Definition.}
|
||||||
|
}{
|
||||||
|
\end{theoremic}
|
||||||
|
}
|
||||||
|
|
||||||
|
\newenvironment{example}{
|
||||||
|
\begin{theoremic}[\definitionfont]{Beispiel.}
|
||||||
|
}{
|
||||||
|
\end{theoremic}
|
||||||
|
}
|
||||||
|
|
||||||
|
|
||||||
Loading…
Add table
Add a link
Reference in a new issue